Open AccessIn this paper we investigate the structure of ring diagrams with periodic labels in groups with small cancellation conditions C(3)-T(6). These diagrams are used to solve the power conjugacy search problem in a cyclic subgroup and the problem of power conjugation. In groups of this class the first problem is positively solved. The second is formulated as follows.
To ascertain whether there are m, n, integers for which the degrees of v, w words with m, n exponents, respectively, are conjugated in the group G = (X; R). To solve this problem in the group with small cancellation conditions C(3)-T(6) it is sufficient to obtain upper bounds for the lengths of the boundary labels of the ring diagram. Therein lies this work.
Previously, it was proved that for any w word we can take a so-called normal form | word with the following property: each degree of normal form is R-irreducible. Studying of the ring diagrams with irreducible periodic labels manages to break a set of these diagrams into three classes. Working with one of these classes and using the periodicity of the boundary labels of diagram, it is possible to prove the periodicity of the layers of this diagram, and further limit the length of the boundary labels.
Thus, it turns out that to solve the power conjugacy problem is enough to use the finite known in advance number of checks for the conjugation of the certain degrees of v, w words. In the considered class of groups this problem is solved long ago.